In Martinis' recent Caltech lecture on the Sycamore paper, he appears to make much of the fact that FIG. 4 of the paper show straight-line fidelity - that is, the fidelity decreases log-linearly with an increase in the number of qubits, and an increase in the depth (number of gates) of the circuit.

I believe a position of Martinis is that the scientific value of Sycamore, which is independent of the claims of supremacy, is that errors can be modeled collectively as digital Pauli errors, and further these errors are independently distributed and don't cluster. This seems to conflict with, for example, Kalai's objection to quantum computing that errors collude and coordinate with increasing qubit count/depth.

Martinis notes that "high school probability" can be used to model the decrease in fidelity with an increase in qubit count/gate count.

But what is this "high school probability" that Martinis refers to?

If I were to model insertion of errors into a large-depth circuit, I might decide, with probability $p$, to insert a randomly selected Pauli error (bit-flip and/or phase shift) between any two gates. The probability would be independent of the depth/qubit count of the circuit.

Is this "Poisson-process" of error-insertion similar to the high school probability to which Martinis refers?


1 Answer 1


The model's accuracy is purely empirical observation. The error trend (Fig 4, or 41:50 in the video) demonstrates that the error of the system (cross entropy fidelity with respect to simulated results) is tracked closely by the "high school probability" model he mentions.

The way this basic model would work is to assume 1- and 2-qubit gate errors are Markovian (time-independent) and also occur independently of other qubits' behavior and then take the product of all the fidelities of gates acting on a single qubit as its total error at measurement time. So if I have the circuit $X(q_0) CNOT(q_0, q_1) X(q_0)$ with an X-gate fidelity of 95% and a CNOT fidelity of 90%, I might say the total error (before applying measurement error) on $q_0$ is $.95 \times .90 \times .95 = 82\%$ and the error on $q_1$ is $90\%$. This probably isn't the exact way they implemented this but it captures how you can might predict the fidelity of the full circuit assuming that qubit errors occur independently.

This seems to conflict with, for example, Kalai's objection to quantum computing that errors collude and coordinate with increasing qubit count/depth.

That objection is probably still valid in general. Since this model's performance was an empirical observation using the results of a specific family of circuits, the validity of such a model won't necessarily generalize to more complicated circuits. Its likely that there will be types of circuits for which correlated errors contribute significantly to the output accuracy.

  • $\begingroup$ Thanks! I read "Markovian" (time-independent with respect to the firing of microwave pulses) as "Poissonian" (space-independent with respect to adding a a bit flip/phase shift to the written-out circuit diagram). As for "types of circuits for which correlated errors contribute" - the Sycamore gate set was universal, right? So even if there are other circuits with correlated errors, if they are Markovian on a universal gate set, then why not use the universal gate set instead? $\endgroup$ Nov 4, 2019 at 18:12
  • $\begingroup$ here I'm using "Markovian" to refer to the noise model applied to the qubit - basically the noise channels are parameterized by time-independent values. I think this is different than the model you're describing. $\endgroup$
    – forky40
    Nov 4, 2019 at 20:11
  • $\begingroup$ re: correlated errors, its more about the type of circuit than the gateset. It is probably a special case that errors in the outcomes they were calculating from the random circuits (Porter-Thomas distribution etc.) could be modeled using 1- and 2-qubit gate errors (plus some extra) only. I imagine that circuits involving preparation and manipulation of a 20-qubit GHZ state wouldn't behave so nicely (but will be pleasantly surprised if thats the case) $\endgroup$
    – forky40
    Nov 4, 2019 at 20:15
  • $\begingroup$ There are scenarios that might call for non-Markovian noise models. Two examples are qubits coupling to a common fluctuator (TLS or otherwise) and $1/f^\alpha$ noise (1-over-f noise is a huge topic in SC qubits). White noise = Markovian only occurs if the spectral noise is flat, which is fairly unphysical $\endgroup$
    – forky40
    Nov 4, 2019 at 23:21
  • $\begingroup$ re: "In other words you would just march down the line of FIG. 4 to some ridiculously low fidelity" - well, that would be great! My argument here is that the burden of evidence is on Google to show that this is the case. Since its expected that a complete noise model is necessary to model arbitrary hardware circuit outcomes, it will require more empirical results to demonstrate that a independent qubit error model is sufficient for predicting general hardware behavior. $\endgroup$
    – forky40
    Nov 4, 2019 at 23:26

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